In a Dedekind complete Riesz space, E, we show that if \((P_n)\) is a sequence of band projections in E then$$\begin{aligned} \limsup \limits _{n\rightarrow \infty } P_n - \liminf \limits _{n\rightarrow \infty } P_n = \limsup \limits _{n\rightarrow \infty } P_n(I-P_{n+1}). \end{aligned}$$This identity is used to obtain conditional extensions in a Dedekind complete Riesz spaces with weak order unit and conditional expectation operator of the Barndorff-Nielsen and Balakrishnan–Stepanov generalizations of the first Borel–Cantelli theorem.

中文翻译：

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Barndorff-Nielsen和Balakrishnan–Stepanov定理的推广到Riesz空间

在Dedekind完全Riesz空间E中，我们证明如果\（（P_n）\）是E中的带投影序列，则$$ \ begin {aligned} \ limsup \ limits _ {n \ rightarrow \ infty} P_n- \ liminf \ limits _ {n \ rightarrow \ infty} P_n = \ limsup \ limits _ {n \ rightarrow \ infty} P_n（I-P_ {n + 1}）。\ end {aligned} $$此身份用于获得带弱阶单位的Dedekind完整Riesz空间中的条件扩展，以及第一个Borel-Cantelli定理的Barndorff-Nielsen和Balakrishnan-Stepanov推广的条件期望算符。